The proof of Theorem REF  is a consequence of several results
proved in Section REF . In particular the basic axioms for
a compact spectral triple (according to [1]}) are proved in
Propositions REF  and REF . The
graded structure provided by \(\chi \)  shows that the spectral triple
is even, which is a manifestation of the dimension of the NGC manifold described by the Dirac operator. The regularity property
defined, as defined in [1]}, is proved in
Proposition REF . We will refer to \(({S}_B,{H}_4,D_B)\)  as
the magnetic spectral triple or the magnetic \(K\) -cycle. We
also note that our magnetic spectral triple satisfies all the axioms for a
noncommutative spin geometry, with the exception of the existence of a real
structure, in a similar fashion to the spectral triple defined
in [3]}.
